The change of variable factor is the absolute value of the determinant
\begin{align*}
\left| \jacm{\cvarf}(\rho,\theta,\phi) \right| = \rho^2 \sin\phi.
\end{align*}
This means that the map from spherical coordinates to rectangular
coordinates changes volume by the factor $\rho^2 \sin\phi$. For this
reason, you need to do the above calculation only once. Now, you can
just remember that the factor for spherical coordinates is $\rho^2 \sin\phi$.

Now we're ready for the example: find the mass of a star $\dlv$ that is a
ball of radius 3 centered at the origin if the density of the star
is $g(x,y,z) = 10 - x^2-y^2-z^2$.

If we try to compute the integral directly in rectangular coordinates,
it isn't so easy:
\begin{align*}
&\iiint_\dlv g(x,y,z) dx\,dy\,dz\\
&= \int_{-3}^3 \int_{-\sqrt{9-z^2}}^{\sqrt{9-z^2}}
\int_{-\sqrt{9-z^2-y^2}}^{\sqrt{9-z^2-y^2}} (10 - x^2
-y^2-z^2)
dx\,dy\,dz
\end{align*}
This would lead to a mess.

We can find the mass of the star more easily in spherical coordinates.
The star density $g(x,y,z) = 10 - x^2-y^2-z^2$
becomes $g(\cvarf(\rho,\theta,\phi)) = 10-\rho^2$.

Applet loading

On the triple integral examples page, we tried to find the volume of an ice cream cone $\dlv$
and discovered the volume was
\begin{align*}
\iiint_\dlv dV
= \int_{-1/\sqrt{2}}^{1/\sqrt{2}} \int_{-\sqrt{1/2-x^2}}^{\sqrt{1/2-x^2}}
\int_{\sqrt{x^2+y^2}}^{\sqrt{1-x^2-y^2}}
dz\,dy\,dx.
\end{align*}

We need to describe the bounds in terms of spherical coordinates.
Since the cone is symmetrical around the $z$ axis, $\theta$ is easy.
$0 \le \theta \le 2\pi$ in the cone and the ranges of the other
variables don't depend on $\theta$. Also, we see that $0 \le r \le
1$, since a given line from the origin extends until it hits the sphere
$z=\sqrt{1-x^2-y^2}$, which is a sphere of radius 1. Lastly, $\phi$
is determined by requiring that we are in the cone $z \ge
\sqrt{x^2+y^2}$. (To see this is exactly a condition on $\phi$, look
at the page on spherical coordinates.) On the cone,
$z=\sqrt{x^2+y^2}$, $\phi = \pi/4$. Consquently, the condition on
$\phi$ is $0 \le \phi \le \pi/4$.